Temperature and Heat — Question 9

Question 9

A cooling law turns temperature history into transferred heat

A body of constant heat capacity $C=600\,\mathrm{J/K}$ cools in a 290 K bath. Its measured temperature is $T(t)=290+30e^{-t/120}$ K, with time in seconds. There is no work and $dQ=C\,dT$, positive inward.

1. Find the instantaneous signed heat-transfer rate and total heat transferred during the first 120 s.

2. Find the time by which 90% of the total eventual outward heat has left the body.

Question 9 - Solution

A cooling law turns temperature history into transferred heat

(a) Method and calculation

$$\dot Q=C\frac{dT}{dt}=-150e^{-t/120}\,\mathrm W.$$$$Q_{0\to120}=600[30e^{-1}-30]\approx-11378.17\,\mathrm J.$$

(b) Interpretation and further calculation

$$Q_{\mathrm{out,\infty}}=600(30)=18000\,\mathrm J.$$$$\frac{Q_{\mathrm{out}}(t)}{18000}=1-e^{-t/120}=.90\ \Rightarrow\ t=120\ln10\approx276.31\,\mathrm s.$$The remaining temperature excess is then 3.00 K.

Physical check

Outward transfer has negative signed heat; its rate tends to zero as the temperature difference vanishes.